Why can't I multiply independent binomial probabilities for drawing colored M&Ms?

[OneSquared]

I have a bag of M&M's that is 22.5% Blue, 12.5%Brown, and 65% other.

If I pull 12 M&M's from the bag, what is the probabiliity that exactly 2 are blue and 3 are brown?

I used the binomial to find the probability of 2 blue and 3 brown, and I want to multiply them together to get the answer, but wouldn't that assume that the two are independent? Obviously they are not, because any time I pull out a blue M&M, it is one time I have not pulled out a brown M&M.

Can someone shed light on how to solve this? Thanks!

 

[juvenal]

There is no replacement, so you can't use a multinomial distribution. You want a multivariate hypergeometric distribution. I found a formula on the web for it here:

http://www.agner.org/random/distrib.pdf

 

[juvenal]

Correction - if you assume an infinite number of M&M's in the bag, then it's probably safe to use the multinomial distribution. The formula for that is also in the link I provided.